How do you normally say this?

Most English-natives do not use "blocked" to discuss their noses unless they want to be very technical--such as "My nasal passages are blocked." The common phrase is, "I have a stuffy nose," or "My nose is runny."

One uses "runny" instead of "running" because the nose isn't going somewhere, it's just full of a viscous substance. If you had a "running nose," your nose would technically be getting up off your face and jogging around the room. If you have a "runny nose," on the other hand, your nose is clogged and starting to drip stuff. If you want, you can say "My nose is running," but this is less common except if you're asked something such as, "What's that on your face?" and you just reached up and found it was mucus from your nose.

You say my nose is blocked
You say I have a runny nose

You can use the sentence "My nose is blocked", if it is completely obstructed.
Usually people say "My nose is stuffed up", or "My nose is stuffy".
Also instead of, "I have a running nose", it is "I have a runny nose".

Of the first two, either are correct and neither more common. And "I have a runny nose" is a common way of saying it.

My nose is running.
My nose is stuffed.